The finite pole s and zeros of a rational matrix G(s) are defined to be the zeros of the polynomial denominator and numerator matrices respectively taken from any relatively prime matrix fraction description of G(s). The infinite poles and zeros of G(s) are then defined as the poles and zeros at w = O of G (1/w). This new definition is the central result of the thesis. From it various results for the theory of rational matrices are derived, many of which are analagous to results in the complex variable theory of rational functions. The infinite pole s and zeros of a rational matrix are investigated in particular and the definition and results obtained are compared and contrasted with alternative definitions of the infinite poles and zero s as described by other authors.

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